Lesson video

Hi, my name is Mr. Peters, and welcome to today's lesson.

In this lesson, we're gonna be extending our thinking around our understanding of factors, and we're gonna think about how we can find all of the factors of a number in a systematic way.

When you're ready to get started, let's get going.

So by the end of this lesson today, you should be able to say that I can find all of the factors of a number in a systematic way.

Throughout this lesson today, we've got two key words we're gonna be referring to.

I'll say them first, and then you can repeat them after me.

The first one is factor, your turn.

The second one is systematic, your turn.

So thinking about what these mean, factors are whole numbers, which exactly divide another whole number.

And being systematic means working in a set way, often according to a fixed plan.

In this lesson today, we've got two cycles we're gonna be working through.

The first one is gonna be finding factors of a number, and the second one is gonna be working systematically to find factors.

Let's get started with the first cycle.

Throughout this lesson, you're gonna meet Sofia and Aisha.

They've got a game they're gonna be playing, as well as they're gonna be sharing some of their thinking throughout the whole of the lesson.

Let's get going.

So Aisha and Sofia are playing a game, and this game's called Fill 200.

Hopefully, you can see there's a grid of 200 squares on the screen.

Aisha and Sofia are gonna run through the rules of the game.

So firstly, they need to pick a card from the deck.

As you can see, Aisha's picked number six.

Then they have to shade in a rectangle, which represents that six onto the grid, as you can see here.

Sofia then gets to have a go, she's picked a number, number 16.

When Sofia was thinking about the factors that would make 16, she's realised that she'd be able to make a square with the number 16, with four rows and four columns.

So therefore, she's decided to shade in a square, because if you're able to make a square, you get another go straight away.

So as you can see, Sofia has picked number 23 now.

Now, 23 only has factors of one and itself, therefore it can only be placed in one long row.

As a result of that, you can see it doesn't fit onto the grid.

So if you get a number which you can't place onto the grid, that means you then lose your turn and don't get to put anything onto the grid.

And so, the goes go forward and backwards over and over again, until the person who fills in the most squares is the winner.

So they've both had a go, let's start from here then.

Aisha pulls number 15 out of the pack, she says, I'm not sure I can make a rectangle with number 15.

Sofia disagrees, she thinks she can.

Let's have a quick look, shall we? Aisha's had a go at making a rectangle, and this is what she's made.

This is the closest she's been able to make to get to 15.

Sofia thinks, hmm, I think we could change this slightly.

Let's have a look, what if we move these three tiles to here instead? Now, we can see we've got a rectangle, haven't we? With three rows and five columns.

Sofia says that she knew this, because she knew that three and five were factors of 15.

So Aisha's remembered, doesn't she now? She's gone, of course, we could do that.

And she's also remembered that every number has a factor of one and itself, doesn't it? We could also represent it like this, as one row of 15 columns.

Aisha decided she's gonna use her one row of 15, and she's gonna replace this at the top of the grid.

Now, it's Sofia's turn.

Sofia draws number 20.

She's realised that 20 has quite a lot of factors, so there's quite a few to choose from.

Shall we have a look at what she could possibly use? Sofia says number 20 has factors of two and 10.

That's because 20 is in the two times table, so two is a factor of 20.

20 is in the 10 times table, so 10 is a factor of 20.

And 20 is also in the four and five times table, so four is a factor of 20, and five is a factor of 20.

20 also has factors of one and itself.

So there's lots of different choices here that Sofia could use.

So let's have a look at the shapes that she's chosen.

She could have chosen the one and 20, which would take one long row, wouldn't it? Or she could choose these two here.

She could choose one with two rows and 10 columns, or five rows and four columns.

She's decided she wants to use the five rows and four columns.

So Sofia's gonna place this in, but she's decided to place that one here instead.

Hmm, is that the shape she had originally? It was the same shape, it's just been rotated round, hasn't it? So instead of having five rows and four columns, it actually has four rows and five columns.

And we know that still is gonna leave us with the same amount, isn't it, altogether? Now, it's Aisha's turn again.

She's drawn 24.

24's in a lot of times stable, she says.

She's wondering what she could make this time.

I wonder if you could help her to figure that out.

So, tick the times table facts that would help Aisha to find the factors of 24, take a moment to have a think.

That's right, it's two and 12, four and six, or three and eight.

Are there any other choices? Well, of course, we could have one and itself, couldn't we? So that would also be another pair of factors that make 24.

Another quick check, true or false, one is the smallest factor of 24? That's true, isn't it? And us thee justifications blow to help reason why.

That's right, it would be A, wouldn't it? One is a factor of every whole number, or every positive integer.

Okay, onto your first task for today then.

What I'll let you do is carry on playing the game.

You can play with a partner if you can.

If not, you can have double the amount of goes, and see how many different shapes you can shade in on the grid.

Good luck, and I'll see you again shortly.

Okay, welcome back.

Your game may have played out in lots and lots of different ways, wasn't it? And here you can see the yellow colours, which represented what Aisha coloured in.

And the red colours represent the boxes that Sofia shaded in.

Who'd you think won? It looks like Sofia's won.

They've run outta cards to play with and therefore, the winner was the person with the most shaded in at that point, which was Sofia.

Well done, you two.

Great game.

Okay, let's move on to cycle two of our lesson now, working systematically to find the factors of a number.

Aisha's saying, if she knew a way to find all of the factors of a number really quickly, that might have helped her to be a bit more successful in the game.

She might have had more choices as to what she could have done when shading in the boxes on the grid.

Sofia's gonna show Aisha how she finds all of the factors really quickly.

Let's look at number 18, and we're gonna use a factor bug to help represent the factors.

Here is the head of the factor bug, and we've placed the number 18 in, because that's the number we're gonna find the factors of.

We know that every number has one and itself as factors, so we're gonna place these as the antennae on the bug.

Then we need to start thinking about, what would be the best set of factors to start looking for next? Aisha says, "I think six is a factor." And she'd be right, and Sofia's pointing that out.

She says, six is a factor of 18, but we've jumped here.

We found factors of one and 18, and actually, that's quite a jump from one up to six, isn't it? We might have missed quite a few different factors along the way.

So maybe working up a number at a time might be the best way.

We've already found the factor pair that works with one, so let's find a factor pair that works with two if we can.

By working up a number at a time, we're going to be starting to work more systematically.

So let's have a think, is 18 in the two times table? Yes, we know that 18 is in the two times table.

We know that two multiplied by nine is equal to 18, so therefore, two and nine would be factors of 18.

Here's the body of the bug now, and we're gonna draw our factors on to either side using legs.

Now we've done one and now we've done two, what would be next to try? That's right, Aisha, we should try three now, shouldn't we? So is 18 in the three times table? We know that there are six threes in 18, don't we? So that's another pair of factors there, isn't it? Six and three are factors of 18, or 18 is the product of six and three.

So let's put three and six onto our bug as well.

There we go.

Okay, is there anything else we could try next? Well, Aisha's saying we should try four next, shouldn't we? Let's give it a go.

In order to check if 18's in the four times table, Aisha's beginning to skip count.

She said four, eight 12, 16, 20.

No it's not, she's realised that 18 isn't in the four times table.

So what about five then? Let's try that.

Is 18 in the five times table? No, it's not, is it? Aisha knows that any number in the five times table have to end in either a five or a zero.

And 18 doesn't end in a five or a zero, so it can't be in the five times table either.

Sofia's starting to think, do we need to check the next number? Well, actually we know we don't need to, do we? 'Cause we can already see on our factor bug that six is a factor and we've already represented that, 'cause it was part of the factor pair with three.

So we don't need to check the next number either.

Sofia has also got a really important piece of information here.

She's realised that we wouldn't need to check anymore either because the quotient is now smaller than the divisor.

Let's have a look at what she means by that.

So we can see here below a division equation, which represents 18 divided by six is equal to three.

This was the last pair of factors that we had, wasn't it? The six and the three as the divisor and the quotient.

And she's realised that the quotient of three is smaller now than the divisor, which is six.

And when the quotient starts to get increasingly smaller than the divisor, we know we've run out of factors.

So we know that we can stop looking for factors at this point, when the quotient is smaller than the divisor.

Looking at that in even more detail, we can see here that 18 divided by one is equal to 18.

We've got one row of 18 to represent that, that's 18 divided into one row is equal to 18.

We can say that 18 divided into two rows is equal to nine, and you can see nine in each row there.

We can see 18 divided into three rows is six, and we've got three rows of six there.

Now, we can see 18 divides into four rows.

Well, here we can see that this hasn't worked equally, has it? And we haven't completed the array here.

So we can see here that 18 divides into four rows, leaves four in each row with a remaining two, doesn't it? Let's have a look at this one, 18 divides into five rows.

Well, again, we can see here 18 divided by five leaves us with three columns, doesn't it? And an additional remainder of three this time.

And then finally, 18 divided into six rows leaves us with those three columns there that are complete and full.

So we can see, as soon as the quotient starts to decrease in size in comparison to the divisor, then we can stop looking for factors.

We can see that 18 divided by six is three, the three is the quotient, and the six was the divisor.

And we know that the quotient is smaller than the divisor here.

And we can see here that three remainder three, as the quotient is smaller than the divisor of five.

Therefore, we didn't even need to search to see if five was a factor of 18.

Okay, quick check for understanding now.

Which factor bug is correct A, B, or C? Take a moment to have a think.

That's right.

It's B, isn't it? We're looking for factors of 12, and we know that the factors of 12 are one and 12, two and six, and three and four.

Well done if you got that.

Another quick check, why are there no factors between six and 12? Ah, that's right, isn't it? Remembering our rule about the quotient being smaller than the divisor.

If you divide 12 by six or greater, then the quotient will always be larger than the divisor.

And therefore, that's the reason for why there's no factors between six and 12.

Sofia is just securing that understanding with an example here.

She says, if you divide 12 by six, that leaves us with two, and then two as a quotient is smaller than the divisor.

At this point, we should have already realised that we've already got these factors.

We've got two and six as the first set of factors.

So therefore, we don't need to repeat ourselves and rewrite those factors onto the bug either, 'cause we've already placed them on before.

Okay, time for you to have a bit of a practise now then.

Can you systematically find all of the factors for these numbers here, 24, 48, 72? Once you've done that, have a quick look at them, what do you notice about them? Once you've done that, I'd like to have a think about this.

If a number has the factors, two, five, and six, what could the number be? There might be more than one solution, so good luck finding as many as you can.

Have a go at those and when you come back, we'll go through them, good luck.

Okay, here are the factors of 24, 48, and 72.

I'll let you have a quick look, and tick them off if you managed to get all of those.

Was there anything in particular that you noticed? Sofia's saying that she's noticed that 48 is twice the size of 24.

And as a result of that, she's noticed that if you take a factor pair of 24, for example, one and 24, if we're doubling the product that we're looking at, i.

e.

, we're changing 24 to 48, then one of the factors in the pair could also be doubled as well.

So the factor pair for 24 was one and 24, and then the factor pair for 48 was one and 48.

So the 24 has been doubled to make 48, just like the original number had.

Let's have a look at another example of that.

24 has the factor pair of two and 12.

48 has the factor pair of two and 24.

48 is double 24.

So if we take one of the factor pairs from 24, two and 12, and double one of them, so let's double the 12, that makes 24.

So we could say that a factor pair of 48 would be two and 24.

Because the product has been doubled, we need to double one of the factors in the pair as well.

You may have also noticed that 72 is three times the size of 24.

So if we use the factor pair again of two and 12, we can take one of those factors, and we can multiply that by three to make it three times the size as well.

Because the product has been made three times the size, we can make one of the factors three times the size and we'll still get the correct factors.

So if two and 12 were factors of 24, then we can take the 12 and multiply that by three, which gives us 36.

So two and 36 would also be a factor pair, this time of 72.

Well done if you managed to spot that.

Okay, and onto the second task now then.

A number has a factor of two, five, and six, what could the possible numbers be? Sofia says for a number to have factors two and five, that must mean it's an even number in the five times table.

That's because two multiplied by five is equal to 10, and that ends in a zero.

So any other number that we multiply by two and five will always end in a zero, 'cause it'll be a multiple of 10.

We could multiply it by six, which would leave us with 60.

So 60 could be a possible solution for this.

However, there's also a number before there, isn't there? Hmm, let's have a think.

Sofia's saying that the first number divisible in the five times table by six would be 30.

30 is also divisible by six, so the answer could be 30 as well.

Let's have a look at our factor bug representing the factors of 30.

30 has factors of one and 30, one and itself, two and 15, three and 10, and five and six.

Aisha's, saying it could have also been 60, which we've already mentioned, and it could also have been 90.

All of those are multiples of 30 and therefore, they are also multiples of two, five, and six.

Well done if you managed to get that.

Okay, that's the end of our lesson for today.

Hopefully you've enjoyed yourself, and you know how to work now more systematically to find the factors of any number.

To summarise our learning, factors can be listed systematically using factor bugs to help ensure that we find all of the factors.

And you can stop searching for factors when you begin to identify factors that you have already found, or when the quotient is smaller than the divisor.

Thanks for joining me today.

Take care, and I'll see you again soon.